Failure of Ornstein-Zernike asymptotics for models with exponentially decaying interactions

VIENNA PROBABILITY SEMINAR

In 1914 and 1916, Ornstein and Zernike published a celebrated heuristic predicting the sharp asymptotic behaviour of density-density correlations in high temperature/low density gases. This theory has since then become a classical topic in the statistical theory of fluids. The (non-rigorous) derivation of their asymptotics relies on a certain "mass gap condition", which was believed to hold at least in very high temperature/very low density regimes as soon as the potential is "short range" (exponentially decaying). In a recent joint work with Y. Aoun, D. Ioffe, and Y. Velenik, we proved that this condition can fail precisely in these regimes, when the interaction decays exactly like $psi(r) e^{-r}$. The failure/success of the condition (and of the OZ asymptotics) is then closely related to the pre-factor psi. In this talk, I will present the mechanism behind this failure, which can be seen as a condensation transition for a suitable model. To keep things elementary, I will mostly restrict to the case of the Ising model.